Preconditioned explicit block simultaneous iterative methods

In this paper, the concept of ‘Preconditioning’ to accelerate the convergence rate of the explicit 4 point block iterative methods is examined. Experimental results for the 1^st and 2^nd order Simultaneous Displacement Methods for the explicit 4 point block schemes indicate that the preconditioning strategy yields substantial gains over the point iterative methods.     

Modulation of Endocannabinoids by Caloric Restriction Is Conserved in Mice but Is Not Required for Protection from Acute Kidney Injury

Acute kidney injury (AKI) is a frequent and critical complication in the clinical setting. In rodents, AKI can be effectively prevented through caloric restriction (CR), which has also been shown to increase lifespan in many species. In Caenorhabditis elegans (C. elegans), longevity studies revealed that a marked CR-induced reduction of endocannabinoids may be a key mechanism. Thus, we hypothesized that regulation of endocannabinoids, particularly arachidonoyl ethanolamide (AEA), might also play a role in CR-mediated protection from renal ischemia-reperfusion injury (IRI) in mammals including humans. In male C57Bl6J mice, CR significantly reduced renal IRI and led to a significant decrease of AEA. Supplementation of AEA to near-normal serum concentrations by repetitive intraperitoneal administration in CR mice, however, did not abrogate the protective effect of CR. We also analyzed serum samples taken before and after CR from patients of three different pilot trials of dietary interventions. In contrast to mice and C. elegans, we detected an increase of AEA. We conclude that endocannabinoid levels in mice are modulated by CR, but CR-mediated renal protection does not depend on this effect. Moreover, our results indicate that modulation of endocannabinoids by CR in humans may differ fundamentally from the effects in animal models.     

Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling

The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three‐dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short‐term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright © 2006 John Wiley & Sons, Ltd.     

Gene control of acupuncture and moxibustion preconditioning on apoptosis in ischemic cardiac muscle of rats with re-perfusion

In order to explore the effect of acupuncture preconditioning on rats' cell apoptosis with cardiac muscle re-perfusion damage and bcl-2mRNA genes, we used differentiating acupuncture and moxibustion preconditioning among groups, then compared acupuncture and moxibustion preconditioning with ischemic preconditioning. The experimental results show that acupuncture and moxibustion preconditioning makes more bcl-2mRNA genes expressed and produces less cell apoptosis, furthermore, groups of acupuncture and moxibustion preconditioning for twice a day are more effective than those of ischemic preconditioning.     

Preconditioning strategies for models of incompressible flow

We describe some new preconditioning strategies for handling the algebraic systems of equations that arise from discretization of the incompressible Navier-Stokes equations. We demonstrate how these methods adapt in a straightforward manner to decisions on implicit or explicit time discretization, explore their use on a collection of benchmark problems, and show how they relate to classical techniques such as projection methods and SIMPLE.     

Assessment of three preconditioning schemes for solution of the two‐dimensional Euler equations at low Mach number flows

Three preconditioners proposed by Eriksson, Choi and Merkel, and Turkel are implemented in a 2D upwind Euler flow solver on unstructured meshes. The mathematical formulations of these preconditioning schemes for different sets of primitive variables are drawn, and their eigenvalues and eigenvectors are compared with each other. For this purpose, these preconditioning schemes are expressed in a unified formulation. A cell‐centered finite volume Roe's method is used for the discretization of the preconditioned Euler equations. The accuracy and performance of these preconditioning schemes are examined by computing steady low Mach number flows over a NACA0012 airfoil and a two‐element NACA4412–4415 airfoil for different conditions. The study shows that these preconditioning schemes greatly enhance the accuracy and convergence rate of the solution of low Mach number flows. The study indicates that the preconditioning methods implemented provide nearly the same results in accuracy; however, they give different performances in convergence rate. It is demonstrated that although the convergence rate of steady solutions is almost independent of the choice of primitive variables and the structure of eigenvectors and their orthogonality, the condition number of the system of equations plays an important role, and it determines the convergence characteristics of solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

散射问题中复线性系统的扰动预条件技术

利用稀疏策略可以控制不完全分解因子的稀疏度,对角扰动技术则通过对原系数矩阵的对角元的轻微扰动,提高不完全分解预条件方法的效率.本文结合稀疏策略和对角扰动技术的修正的不完全LLT分解预条件技术,用来加速共轭垂直共轭梯度法(COCG)求解离散散射问题得到的大型、稀疏的复对称线性系统的求解速率,数值试验验证了基于扰动的不完全分解预条件方法,对迭代求解散射问题有着很好的提速效果.     

A class of approximate inverse preconditioners for solving linear systems

Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587–1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737–1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97–103]. Zhang et al. ’s preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons’.     

Preconditioning of the second‐kind boundary integral equations for 3D eddy current problems

Starting from the time‐harmonic Maxwell equations at low‐frequency eddy current approximation the H–? formulation is presented. An equivalent system of boundary integral equations of the second kind on the conductor surface (resp. the conductor/dielectric) is derived. Discretizing these equations with a boundary element method (BEM) yields a block linear equation system \[ \left[\begin{array}{cc} A_{1} & B_{1} \\ \tfrac{\mu}{\mu_{0}}\tilde{B}_{2} & A_{2}\end{array} \right] \left( \begin{array}{c} j \\ \sigma \end{array} \right)=\left(\begin{array}{c} b_{1} \\ b_{2}\end{array} \right) \] with a fully populated stiffness matrix. This system is non‐symmetric and, for large μ/μ₀ (of interest in practice) ill‐conditioned. Iterative solvers like GMRES converge very slowly, in general. We propose a new preconditioner which depends on μ. We present numerical results with a serial and parallel version of this preconditioner also on large industrial eddy current problems with compex geometry. The performance of preconditioned GMRES is found to be practically independent of μ/μ₀. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical solution of three-dimensional boundary-value problems by generalized approximate inverse matrix techniques

Generalized Approximate Inverse Matrix (GAIM) techniques based on the concept of LU-type sparse factorization procedures are introduced for calculating explicitly approximate inverses of large sparse unsymmetric matrices of regular structure without inverting the factors L and U. Explicit first and second-order iterative methods in conjunction with modified forms of the GAIM techniques are presented for solving numerically three-dimensional initial/boundary-value problems on multiprocessor systems. Applications of the new methods on a 3D boundary-value problem is discussed and numerical results are given.